A new energy method for the Boltzmann equation
Tong Yang
a͒
Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong
Hui-Jiang Zhao
b͒
School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China
͑Received 31 May 2005; accepted 21 March 2006; published online 4 May 2006
͒
An energy method for the Boltzmann equation was proposed by Liu, Yang, and Yu
͓Physica D 188, 178–192 ͑2004͔͒ based on the decomposition of the Boltzmann
equation and its solution around the local Maxwellian. The main idea is to rewrite
the Boltzmann equation as a fluid-type dynamics system with the nonfluid compo-
nent appearing in the source terms, coupled with an equation for the time evolution
of the nonfluid component. In this paper, we will elaborate this method and our
main observation is that the microscopic projection of the local Maxwellian with
respect to a given global Maxwellian is not linear but quadratic. Based on this and
by analyzing the fluid-type system using the analytic techniques for the system of
conservation laws, we can indeed control the conserved quantities
,
u, and
͑
1
2
u
2
+ E
͒
of the Boltzmann equation by the microscopic projection of the solution
of the Boltzmann equation with respect to the global Maxwellian, which is suffi-
cient to deduce the energy estimates for the solution of the Boltzmann equation.
The main purpose here is to show that there is no need to perform two sets of
energy estimates with respect to the local and a global Maxwellian as in the pre-
vious works. In fact, one set of energy estimates with respect to the global Max-
wellian is sufficient for closing the energy estimates. Therefore, it not only simpli-
fies the analysis in the previous works, but also shed some light on the stability
analysis in some complicated systems, such as the Vlasov-Poisson-Boltzmann and
Vlasov-Maxwell-Boltzmann systems. © 2006 American Institute of Physics.
͓DOI: 10.1063/1.2195528͔
I. INTRODUCTION
Consider the Boltzmann equation
f
t
+
· ٌ
x
f =
1
Q͑f, f͒, ͑f,t,x,
͒ R
+
ϫ R
+
ϫ R
3
ϫ R
3
, ͑1.1͒
where f = f͑t , x ,
͒ represents the distributional density of particles at space–time ͑x ,t͒ with veloc-
ity
,
is the Knudsen number proportional to the mean free path, and Q͑f , f͒ is the collision
operator given by the following bilinear form, cf. Ref. 1,
a͒
Electronic mail: matyang@math.cityu.edu.hk. Research supported by the RGC Competitive Earmarked Research Grant,
CityU 103004.
b͒
Electronic mail: hhjjzhao@hotmail.com. Research supported by the National Natural Science Foundation of China under
Contracts Nos. 10329101 and 10431060, respectively.
JOURNAL OF MATHEMATICAL PHYSICS 47, 053301 ͑2006͒
47, 053301-10022-2488/2006/47͑5͒/053301/19/$23.00 © 2006 American Institute of Physics